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I am asked the following question:

You are responsible for building a highway that connects A to B. There's an old highway 50 miles south that can be restored at the cost of \$300.000,00 per mile. Building a new path requires \$500.000,00 per mile. Find the optimal path that minimizes the cost.

problem

So we can call the projections of A and B A' and B', and the midpoints between them X and Y.

enter image description here

Using a straight path from A to B would require \$75.000.000,00 and going AA'B'B would require \$95.000.000,00. Sonlet's find an optimal point on the middle.

The cost function is given by

$$ C = 2 \cdot 500000 \sqrt{50^2+x^2} + 300000(150-2x)\\ C' = \frac{10x}{\sqrt{x^2+2500}}-6=0 $$

If we say $C'=0$ we don't find any critical points. Did I do a miscalculation? And if not, what is the conclusion of the problem?

Thank you.

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Your equation $C'=0$ does have a solution. Add $6$ to both sides, multiply by $\sqrt{x^2+2500}$, then square both sides, getting $$ 100x^2 = 36(x^2+2500) $$ which has the solution $x=300/8$.

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  • $\begingroup$ Sure I just simplified it. $\endgroup$ – bru1987 Sep 13 '16 at 17:09
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    $\begingroup$ But you simplified incorrectly: the correct way to simplify would be$$C'=\frac{2x}{\sqrt{2500+x^2}}\cdot5-6=0.$$ $\endgroup$ – Mike Earnest Sep 13 '16 at 17:13
  • $\begingroup$ Youre right I forgot to put the 500 factor, thank you $\endgroup$ – bru1987 Sep 13 '16 at 17:14
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    $\begingroup$ So what I have (using the root $75/2$ is a cost of \$85 million, but that is more costly than just building a highway straight from A to B (which would cost \$75 million) right? $\endgroup$ – bru1987 Sep 13 '16 at 18:31
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    $\begingroup$ @bru1987 Correct. $\endgroup$ – dxiv Sep 13 '16 at 19:44

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